Simplify the following expression and state the condition under which the simplification is valid: $t = \dfrac{z^2 - z - 12}{z^2 - 9}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{z^2 - z - 12}{z^2 - 9} = \dfrac{(z - 4)(z + 3)}{(z - 3)(z + 3)} $ Notice that the term $(z + 3)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(z + 3)$ gives: $t = \dfrac{z - 4}{z - 3}$ Since we divided by $(z + 3)$, $z \neq -3$. $t = \dfrac{z - 4}{z - 3}; \space z \neq -3$